Parabola Equation | Standard Forms, Vertex Form & Examples

Parabola Equation

A parabola is a U-shaped curve that can open upward or downward. In coordinate geometry, a parabola is commonly the graph of a quadratic function. This page explains the most-used parabola equations and how to work with them.

📌 Parabola equation (most common)

The most common parabola is a quadratic in the form:

y = ax² + bx + c

  • a controls whether it opens up (a > 0) or down (a < 0), and how “wide” it is.
  • b affects the horizontal placement.
  • c is the y-intercept (where it crosses the y-axis).

Vertex form (best for the turning point)

If you want the turning point (vertex) quickly, use vertex form:

y = a(x − h)² + k

  • The vertex is (h, k).
  • The axis of symmetry is x = h.
  • Opens up if a > 0, opens down if a < 0.

How to find the vertex from y = ax² + bx + c

For y = ax² + bx + c, the x-coordinate of the vertex is:

x = −b / (2a)

Then substitute that x-value back into the equation to find the y-value.

Intercepts

Y-intercept

Set x = 0. Then y = c. So the y-intercept is (0, c).

X-intercepts (roots)

Set y = 0 and solve:

0 = ax² + bx + c

You can solve by factoring (if possible) or the quadratic formula:

x = (−b ± √(b² − 4ac)) / (2a)

🧮 Example 1: Standard form

Given y = x² − 4x + 1:

  • a = 1 so it opens upward.
  • y-intercept: (0, 1)
  • vertex x: x = −(−4)/(2·1) = 4/2 = 2
  • vertex y: y = 2² − 4(2) + 1 = 4 − 8 + 1 = −3

So the vertex is (2, −3) and the axis of symmetry is x = 2.

🧮 Example 2: Convert to vertex form (complete the square)

Convert y = x² − 6x + 5 to vertex form.

  1. Group x terms: y = (x² − 6x) + 5
  2. Complete the square: half of −6 is −3, square it to get 9
  3. Add and subtract 9: y = (x² − 6x + 9) + 5 − 9
  4. Write as a square: y = (x − 3)² − 4

So y = (x − 3)² − 4, with vertex (3, −4).

Quick facts (easy checks)

  • Wider vs narrower: if |a| is small (e.g. 0.5) the parabola is wider; if |a| is large (e.g. 3) it’s narrower.
  • Maximum/minimum: vertex is the minimum if a > 0; maximum if a < 0.
  • Symmetry: points the same distance left/right of x = h have the same y value.

Related topics

  • Quadratic formula
  • Completing the square
  • Discriminant (b² − 4ac)
  • Graphing quadratics

Disclaimer: This page is provided for general educational reference only. While care has been taken to present accurate formulas and examples, results may vary depending on rounding and input precision. This content does not replace professional educational, engineering, or technical advice.